In computer science, the topological form of a target graph is the linear arrangement of its tracts so that throughout the vertical edge a from vertex a to vertex b, a comes before b in planning. Topological sort is a valid sequence of tasks. It is a graph break where each node is visited only after visiting all its dependencies.

Topological sort has many applications, especially for measurement problems like arc response sets. Topological sort is only possible if the graph is (DAG) which is a direct acyclic graph, it is also possible when the (DAG) has a cut-off part.

**An example of a topological sort**

Weather graph filtering “5 4 2 3 1 0”. There may be more than one topological sort on the graph. For example, another topological layout of the same graph is “4 5 2 3 1 0”. The first vertex in topological filtering is always a 0-level vertex (a vertex with no input ends).

**Topological application**

**There are a few ways to apply the topological type to the work we do to make sure we follow the basic steps.**

- Determining the order of integration tasks be performed in a construction file.
- Through the command system
- Find the cycle on the graph
- By arranging tasks from the given dependencies between tasks.
- Production equipment
- With the deadlock closure of the operating system.
- Data processing

**Steps to find the topological sorting**

• See vertex with in-degree = 0 (no inbound edges)

• Remove all edges from the vertex to the outside (make its out-degree = 0, remove the outgoing edges.)

• Add a vertex to the list representing weather graph filtering.

• Repeat the process until there are no vertices left.

**Topological sorting algorithm**

**Two types of algorithms can be used to determine the topological sequence of any graph, These are:**

**• Kahn algorithm**

**• Depth-first search algorithm (DFS).**

### Kahn algorithm

Kahn’s algorithm works by keeping track of the number of incoming edges in each area (degree). Duplicate: Detects nodes with no incoming edge, i.e., nodes with zero degrees (no dependency). It saves zero nodes by degrees in line and removes them from the original graph.

**The complexity of Kahn’s algorithm**

The complex time of the Kahn type algorithm of the topological O (V + E), in which V and E are the total number of vertices and edges on the graph, respectively.

**Space Complexity**

The required space is O (V). We must create a single archive array for all nodes. This will require an O (V) space.

We must keep the nodes with a degree = 0 in the data structure (stack or line) when subtracting them from the graph. In the worst-case scenario, this should keep all graph nodes, so this will require an O (V) space. Lastly, we need the same members to keep all the notes in an orderly fashion. This will naturally require an O (V) space.

In addition to all three, we come to a complex O (V) complex.

### Advanced search algorithm or (DFS).

Depth-First Search or DFS algorithm is a repetitive algorithm that uses a regression method. It involves performing a thorough search of all nodes by moving forward when possible and by stepping down, if necessary. To visit the next node, open the top node in the stack and press all the nodes closest to the stack. Topological sorting, editing problems, graph cycle detection, and solving problems with just one solution, like a maze or sudoku puzzle, all use advanced search algorithms. Some applications include network analysis, such as determining whether a graph is bipartite.

**Depth-first Search (DFS)**

**There are two types of difficulty in the DFS algorithm.**

• Space Complexity

• Time Complexity

Time Complexity for the depth-first search algorithm When the whole graph is broken, the temporal complexity of DFS is O (V), where V is the number of vertices.

**If the graph data structure is displayed as a close-up list, the following rules apply: **

- Each vertex follows all the adjacent edges. Let’s pretend there are V vertices and E edges on the graph.
- You find all the node neighbors by going through its closest list only once during the queue.
- The total size of the approximate list of all vertices in the graph is directed by E. In this example, the interim weight is O (V) + O (E) = O (V + E).
- Each edge of the indirect graph appears twice. Once at the end of the nearest edge list. The temporary complexity of this case will be O (V) + O (2E) O (V + E).
- If the graph is represented as an adjacency matrix V x V the same members:
- To find all the outgoing vertex edges, you will need to skip the entire V-length line in the matrix.
- Each row in the adjacent matrix corresponds to the node in the graph; each line stores information about the edges from that vertex. As a result, the temporary DFS complex, in this case, is O (V * V) = O. (V2).
- The complexity of the space for an in-depth search algorithm,Because you follow the trajectory of the last vertex visited in the stack, the stack can grow into the size of graph graphs in the worst-case scenario. As a result, the local complex is O. (V).

#### Conclusion

Problems involving topological layout are easy to spot and the algorithm can be deployed quickly. You just have to look at the patterns, such as that any task can be accomplished only after certain tasks have been completed, and make sure the resulting graph is acyclic. Problems using the topological type can only be easily solved and therefore standardized, but many problems can be created when obtaining a topo-order is a small step needed to achieve a final response.

#### Frequently Asked Questions

**Question: What does DSF mean?**

**Answer: **Depth-first search.

**Question: What is DAG?**

**Answer:** DAG refers to a directed acyclic graph.

**Question: How much complexity is the topological sort?**

**Answer:** we have 2 types of difficulties. Complex Time and Space.

**Question: Can topological sorting be in a common graph?**

**Answer: **No, topological sorting can only be in a (DAG).